165 research outputs found
Distributed Order Derivatives and Relaxation Patterns
We consider equations of the form , ,
where , is a distributed order derivative, that is the
Caputo-Dzhrbashyan fractional derivative of order , integrated in
with respect to a positive measure . Such equations are
used for modeling anomalous, non-exponential relaxation processes. In this work
we study asymptotic behavior of solutions of the above equation, depending on
properties of the measure
Fractional Cauchy problems on bounded domains: survey of recent results
In a fractional Cauchy problem, the usual first order time derivative is
replaced by a fractional derivative. This problem was first considered by
\citet{nigmatullin}, and \citet{zaslavsky} in for modeling some
physical phenomena.
The fractional derivative models time delays in a diffusion process. We will
give a survey of the recent results on the fractional Cauchy problem and its
generalizations on bounded domains D\subset \rd obtained in \citet{m-n-v-aop,
mnv-2}. We also study the solutions of fractional Cauchy problem where the
first time derivative is replaced with an infinite sum of fractional
derivatives. We point out a connection to eigenvalue problems for the
fractional time operators considered. The solutions to the eigenvalue problems
are expressed by Mittag-Leffler functions and its generalized versions. The
stochastic solution of the eigenvalue problems for the fractional derivatives
are given by inverse subordinators
p-Adic description of characteristic relaxation in complex systems
This work is a further development of an approach to the description of
relaxation processes in complex systems on the basis of the p-adic analysis. We
show that three types of relaxation fitted into the Kohlrausch-Williams-Watts
law, the power decay law, or the logarithmic decay law, are similar random
processes. Inherently, these processes are ultrametric and are described by the
p-adic master equation. The physical meaning of this equation is explained in
terms of a random walk constrained by a hierarchical energy landscape. We also
discuss relations between the relaxation kinetics and the energy landscapes.Comment: AMS-LaTeX (+iopart style), 9 pages, submitted to J.Phys.
p-Adic Mathematical Physics
A brief review of some selected topics in p-adic mathematical physics is
presented.Comment: 36 page
Boundary Conditions for Singular Perturbations of Self-Adjoint Operators
Let A:D(A)\subseteq\H\to\H be an injective self-adjoint operator and let
\tau:D(A)\to\X, X a Banach space, be a surjective linear map such that
\|\tau\phi\|_\X\le c \|A\phi\|_\H. Supposing that \text{\rm Range}
(\tau')\cap\H' =\{0\}, we define a family of self-adjoint
operators which are extensions of the symmetric operator .
Any in the operator domain is characterized by a sort
of boundary conditions on its univocally defined regular component \phireg,
which belongs to the completion of D(A) w.r.t. the norm \|A\phi\|_\H. These
boundary conditions are written in terms of the map , playing the role of
a trace (restriction) operator, as \tau\phireg=\Theta Q_\phi, the extension
parameter being a self-adjoint operator from X' to X. The self-adjoint
extension is then simply defined by A^\tau_\Theta\phi:=A \phireg. The case in
which is a convolution operator on LD, T a distribution with
compact support, is studied in detail.Comment: Revised version. To appear in Operator Theory: Advances and
Applications, vol. 13
Connection Conditions and the Spectral Family under Singular Potentials
To describe a quantum system whose potential is divergent at one point, one
must provide proper connection conditions for the wave functions at the
singularity. Generalizing the scheme used for point interactions in one
dimension, we present a set of connection conditions which are well-defined
even if the wave functions and/or their derivatives are divergent at the
singularity. Our generalized scheme covers the entire U(2) family of
quantizations (self-adjoint Hamiltonians) admitted for the singular system. We
use this scheme to examine the spectra of the Coulomb potential and the harmonic oscillator with square inverse potential , and thereby provide a general perspective for these
models which have previously been treated with restrictive connection
conditions resulting in conflicting spectra. We further show that, for any
parity invariant singular potentials , the spectrum is determined
solely by the eigenvalues of the characteristic matrix .Comment: TeX, 18 page
Kirchhoff's Rule for Quantum Wires
In this article we formulate and discuss one particle quantum scattering
theory on an arbitrary finite graph with open ends and where we define the
Hamiltonian to be (minus) the Laplace operator with general boundary conditions
at the vertices. This results in a scattering theory with channels. The
corresponding on-shell S-matrix formed by the reflection and transmission
amplitudes for incoming plane waves of energy is explicitly given in
terms of the boundary conditions and the lengths of the internal lines. It is
shown to be unitary, which may be viewed as the quantum version of Kirchhoff's
law. We exhibit covariance and symmetry properties. It is symmetric if the
boundary conditions are real. Also there is a duality transformation on the set
of boundary conditions and the lengths of the internal lines such that the low
energy behaviour of one theory gives the high energy behaviour of the
transformed theory. Finally we provide a composition rule by which the on-shell
S-matrix of a graph is factorizable in terms of the S-matrices of its
subgraphs. All proofs only use known facts from the theory of self-adjoint
extensions, standard linear algebra, complex function theory and elementary
arguments from the theory of Hermitean symplectic forms.Comment: 40 page
Towards ultrametric theory of turbulence
Relation of ultrametric analysis, wavelet theory and cascade models of
turbulence is discussed. We construct the explicit solutions for the nonlinear
ultrametric integral equation with quadratic nonlinearity. These solutions are
built by means of the recurrent hierarchical procedure which is analogous to
the procedure used for the cascade models of turbulence.Comment: 11 page
First Passage Time Distribution and Number of Returns for Ultrametric Random Walk
In this paper, we consider a homogeneous Markov process \xi(t;\omega) on an
ultrametric space Q_p, with distribution density f(x,t), x in Q_p, t in R_+,
satisfying the ultrametric diffusion equation df(x,t)/dt =-Df(x,t). We
construct and examine a random variable \tau (\omega) that has the meaning the
first passage times. Also, we obtain a formula for the mean number of returns
on the interval (0,t] and give its asymptotic estimates for large t.Comment: 20 page
Some aspects of the -adic analysis and its applications to -adic stochastic processes
In this paper we consider a generalization of analysis on -adic numbers
field to the case of -adic numbers ring. The basic statements, theorems
and formulas of -adic analysis can be used for the case of -adic analysis
without changing. We discuss basic properties of -adic numbers and consider
some properties of -adic integration and -adic Fourier analysis. The
class of infinitely divisible -adic distributions and the class of -adic
stochastic Levi processes were introduced. The special class of -adic CTRW
process and fractional-time -adic random walk as the diffusive limit of it
is considered. We found the asymptotic behavior of the probability measure of
initial distribution support for fractional-time -adic random walk.Comment: 18 page
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